# American Institute of Mathematical Sciences

January  2020, 16(1): 371-386. doi: 10.3934/jimo.2018157

## Analysis of strategic customer behavior in fuzzy queueing systems

 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author.Email address: math_zjc@csu.edu.cn (Jingchuan Zhang)

Received  June 2018 Published  September 2018

Fund Project: This work is partially supported by the National Natural Science Foundation of China (11671404), and the Fundamental Research Funds for the Central Universities of Central South University (2017zzts061, 2017zzts386).

This paper analyzes the optimal and equilibrium strategies in fuzzy Markovian queues where the system parameters are all fuzzy numbers. In this work, tools from both fuzzy logic and queuing theory have been used to investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. By Zadeh's extension principle and $α$-cut approach, we formulate a pair of parametric nonlinear programs to describe the family of crisp strategy. Then the membership functions of the strategies in single and multi-server models are derived. Furthermore, the grated mean integration method is applied to find estimate of the equilibrium strategy in the fuzzy sense. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and a sensitivity analysis is performed, which show the relationship of these strategies and social benefits. Our finding reveals that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the system managers could get more precise information.

Citation: Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
##### References:

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##### References:
The approximate membership function of optimal and equilibrium threshold $n$.
The approximate membership function of optimal and equilibrium arrival rate $\lambda$.
The approximate membership function of optimal social benefit $S$.
Fuzzy trapezoidal value for the input parameters $\tilde{\Lambda}$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
Fuzzy trapezoidal value for the input parameters $\tilde{\mu}$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{R}$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{C}$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
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